He asked her, 'How long have you lived here?

Direct & Indirect speech:
- There can be two basic narratives to describe a statement in English.
- It is important to understand direct & indirect speech.
- In direct speech, "a quoted speech" is used to narrate someone else's speech.

Remember this transformation of verb tense while changing narratives:

Direct Speech/ Reporting speech Vs Indirect speech / Reported speech

Present indefinite - Past indefinite
Present continuous- - Past continuous
Present Perfect -Past perfect
Past indefinite -Past Perfect
Past continuous - Past perfect continuous
Past perfect - Past Perfect
Future Indefinite - Use of modal

যেহেতু এখানে, direct speech, present perfect সুতরাং change of narrative এর সময় এটি Past perfect form অনুসরণ করবে।

অতএব সঠিক বাক্যটি হল: He asked her how long she had lived there.

Source: Cambridge Dictionary Grammar Reference.